Except it is a blatant 'but thou must' and I despise them, specifically their is no REASON to do that quest, yes it advances the plot, but Hawke does not have a reason to do it, it is simply a box you have to tick to move on, rather than feeling like a part of an evolving plot it is a hook to turn you into a bystander and you agree because 'reasons' that are never elaborated, even DA:I did a better job of answering 'why' questions, 'why do I have to fight Corpsefish' Oh right he will murder me for this doohickey, not a great answer, but an answer, one that doesn't leave the protagonist as a railroaded puppet, or rather hides the rails better, that is why it is annoying it feels so forced and she is so obnoxious that not being able to tell her to screw it is really infuriating. If one of your companions had asked, or it had been necessary to gain chantry support for reclaiming the mansion, then well, we have a reason to do it, something to explain why we are working for this stuck up loon.
I don't understand your argument, though.
There is anti-Fereldan sentiment in Kirkwall. There is anti-Qunari sentiment in Kirkwall. Judgey Chantry Sisters and Mothers are the first to predictably fit into the conservative anti-alien Kirkwall population, and not only that, we know the Chantry is an important function in the human societies of Thedas. Kirkwall is a human society. Just as in our own world, it makes sense that there are zealots and fringe factions that are deeply conservative and want to take matters into their own hands. Hence, someone like Sister Petric - now being introduced as a major character into the story, representing this element of Thedas and the reaction to the Qunari. On the opposite side, Hawke has become a prominent citizen of Kirkwall. The definition of a person who has grown into the very fabric of Kirkwall society. A rags to riches refugee. How would Hawke not be a part of this?
That's what a story consists of: characters, setting, plot. Plot includes cause and effect - the effect contingent on the setting.
Setting - Thedas. Kirkwall. Human society that respects chantry, looks up to chantry, think of Catholic church, past religions, Tea Party.
Cahracters - Hawke. Motivation has been limited by Bioware writer. Hawke wants money. Hawke wants to make a life in Kirwall. That means making friends and inevitably becoming invested in their friends. You can't turn away from that motivation.
Sister Petrice. A character with the motivation to expel foreigners from Kirkwall. Makes sense. Conservative element in Thedas, right.
Qunari. The Qunari are watching, the Qunari need the relic, and they also like to project their Qun on other countries as we have seen before. They have an evanglist-type mandate in them.
Plot - Cause: Because Hawke is now invested in Kirkwall, is always up for more coin, and wants to defend her friends which she cares about, it seems reasonable that she would at least be curious about looking into this suspicious zealot, Sister Petrice, who is going to be antagonizing a major invasive force in her city. Effect: Hawke investigates.